Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Profinite ∞-operads
Stockholm University, Faculty of Science, Department of Mathematics.
2022 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 408, article id 108601Article in journal (Refereed) Published
Abstract [en]

We show that a profinite completion functor for (simplicial or topological) operads with good homotopical properties can be constructed as a left Quillen functor from an appropriate model category of ∞-operads to a certain model category of profinite ∞-operads. The construction is based on a notion of lean ∞-operad, and we characterize those ∞-operads weakly equivalent to lean ones in terms of homotopical finiteness properties. Several variants of the construction are also discussed, such as the cases of unital (or closed) ∞-operads and of ∞-categories. 

Place, publisher, year, edition, pages
2022. Vol. 408, article id 108601
Keywords [en]
Dendroidal sets, Infinity-operads, Lean infinity-operads, Profinite completion, Quillen model categories
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-211817DOI: 10.1016/j.aim.2022.108601ISI: 000860763500003Scopus ID: 2-s2.0-85135502790OAI: oai:DiVA.org:su-211817DiVA, id: diva2:1713800
Available from: 2022-11-28 Created: 2022-11-28 Last updated: 2023-03-12Bibliographically approved
In thesis
1. Model categories, pro-categories and functors
Open this publication in new window or tab >>Model categories, pro-categories and functors
2023 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of five papers. The first three are concerned with various model structures on ind- and pro-categories, while the last two are concerned with the homotopy theory of functors.

In Paper I, a general method for constructing simplicial model structures on ind- and pro-categories is described and its basic properties are studied. This method is particularly useful for constructing "profinite" analogues of known model categories. It recovers various known model structures and also constructs many interesting new model structures. 

In Paper II, it is shown that a profinite completion functor for (simplicial or topological) operads with good homotopical properties can be constructed as a left Quillen functor from an appropriate model category of infinity-operads to a certain model category of profinite infinity-operads. The construction of the latter model category is inspired by the method described in Paper I, but there are a few subtle differences that make its construction more involved.

In Paper III, the general method from Paper I is used to give an alternative proof of a result by Arone, Barnea and Schlank. This result states that the stabilization of the category of noncommutative CW-complexes can be modelled as the category of spectral presheaves on a certain category M. The advantage of this alternative proof is that it mainly relies on well-known results on (stable) model categories.

In Paper IV, the question of whether an ordinary functor between enriched categories is equivalent to an enriched functor is addressed. This is done for several types of enrichments: namely when the base of enrichment is (pointed) topological spaces, (pointed) simplicial sets or orthogonal spectra. Simple criteria are obtained under which this question has a positive answer.

In Paper V, the Goodwillie calculus of functors between categories of enriched diagram spaces is described. It is shown that the layers of the Goodwillie tower are classified by certain types of diagrams in spectra, directly generalizing Goodwillie's original classification. Using this classification, an operad structure on the derivatives of the identity functor is constructed that generalizes an operad structure originally constructed by Ching.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2023. p. 43
Keywords
Homotopy theory, Quillen model categories, Pro-categories, Enriched categories, Goodwillie calculus
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-215391 (URN)978-91-8014-232-8 (ISBN)978-91-8014-233-5 (ISBN)
Public defence
2023-05-25, lärosal 4, hus 1, Albano, Albanovägen 28, Stockholm, 13:15 (English)
Opponent
Supervisors
Available from: 2023-05-02 Created: 2023-03-12 Last updated: 2023-03-27Bibliographically approved

Open Access in DiVA

No full text in DiVA

Other links

Publisher's full textScopus

Authority records

Blom, Thomas

Search in DiVA

By author/editor
Blom, Thomas
By organisation
Department of Mathematics
In the same journal
Advances in Mathematics
Mathematics

Search outside of DiVA

GoogleGoogle Scholar

doi
urn-nbn

Altmetric score

doi
urn-nbn
Total: 27 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf